Quantum Computing Stack Exchange
Quantum Computing Stack Exchange

Questions tagged [stabilizer-code]

A stabilizer quantum error-correcting code appends ancilla qubits to qubits that we want to protect. A unitary encoding circuit rotates the global state into a subspace of a larger Hilbert space. This highly entangled, encoded state corrects for local noisy errors.

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Additive Codes: What are they? And how are they related to stabilizer codes?

I recently encountered the term "additive code" in the answer to my question here What are nontrivial examples of stabilizer codes whose codewords have some $\pm i$ coefficients? Implicit in the answer seems to be the claim that additive codes are…
Ian Gershon Teixeira
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Maximum number of Stabilizer Generators?

The Pauli group, $P_n$, is given by $$P_n=\{ \pm 1, \pm i\}\otimes \{ I,\sigma_x,\sigma_y,\sigma_z\}^{\otimes n}$$ Abelian subgroups of this which do not contain the element $(-1)*I$ correspond to a stabilizer group. If there are $r$ generators of…
Quantum spaghettification
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Qutrit Steane Code

There is a well known 5 qubit code $ [[5,1,3]] $ with stabilizer generators $$ XZZXI \\ IXZZX \\ XIXZZ \\ ZXIXZ $$ There is a corresponding $ [[5,1,3]] $ code for qutrits given by \begin{align*} & XZZ^\dagger X^\dagger I \\ & IXZZ^\dagger X^\dagger…
Ian Gershon Teixeira
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Why do stabilizer codes in Nielsen and Chuang have Pauli X and Z matrices?

While reading about quantum stabilizer codes I have noticed that almost all codes I have seen have Pauli X and Z matrices. Is there any specific reason for this? I have been teaching myself about Quantum stabilizer codes through the 10th chapter of…
alpha
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Why stabilizer codes are named additive quantum codes?

I noticed that stabilizer codes and additive quantum codes are equivalent, but why? I am confused at the word "additive" since the operation of stabilizer genertors is multiplication.
Jacey Li
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Are all stabilizer codes real? Or example stabilizer codes with $\pm i$ coefficients

This is a follow-up question to Stabilizer codes and 1,-1 coefficients A lot of well known codes (5 qubit code, 7 qubit Steane code, 9 qubit Shor code) have logical zero and logical one which can be written as (a global scalar times) a linear…
Ian Gershon Teixeira
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What is the projective measurement operator for a stabilizer?

I am learning about the stabilizer. And for a stabilizer $A$, its eigenspace with $\lambda=1$ corresponds to no-error condition while the eigenspace with $\lambda=-1$ corresponds to errored condition. And I read it that we can define the projective…
QubitTy
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How to distinguish logical X vs logical Z in the stabilizer code with $k=1$?

Let's say I have a stabilizer code with $k=1$. I calculate the normalizer and find the values of $A,B$ that are anticommuting and are not in the stabilizer group. How do I know which one to assign to $\bar X$ and which one to $\bar Z$? Would an…
unknown
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Stabilizer error correction condition

I am learning the stabilizer code through Nielsen & Chuang which give the following hypothesis to have correctable error for stabilizer codes: Let S be the stabilizer for a stabilizer code C(S). Suppose ${E_j}$ is a set of operators in $G_n$ such…
Marco Fellous-Asiani
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How to get codewords from parity check matrix?

I'd like to check Knill-Laflamme conditions given a set of possible errors $\mathcal{E} \subset \mathbb{C}^{2^x \times 2^n}$ and a parity check matrix, i.e. find $c_{ab}$ in $\left\langle\psi_i\left|E_a^{\dagger} E_b\right| \psi_j\right\rangle=c_{a…
tomek
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Example stabilizer code which is not $ GF(4) $ linear

In the paper https://arxiv.org/abs/quant-ph/9704043 Eric Rains talks about $ GF(4) $ linear codes and proves some of their properties, for example "many codes of interest (e.g., GF(4)-linear codes) are built out of distance 2 codes." "Since…
Ian Gershon Teixeira
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Prove that if $|\psi\rangle\neq|\phi\rangle$ then ${\rm Stab}(|\psi\rangle)\neq{\rm Stab}(|\phi\rangle)$

While reading a paper by Aaronson and Gottesman, I came across a comment as follows: $$\text{If }|\psi\rangle \neq |\phi\rangle, Stab(|\psi\rangle) \neq Stab(|\phi\rangle),$$ where $|\psi\rangle$ and $|\phi\rangle$ are two arbitrary states, and…
SML0712
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Is it necessary to define logical Pauli in the centralizer of the stabilizer group

By definition, we define the logical Pauli operator as element of the centralizer of the stabilizer group $S$ in $G_n$ (the $n$-Pauli group). It is a definition so why not. My question is: mathematically speaking, is it really necessary ? Is this…
Marco Fellous-Asiani
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In stabilizer codes, I am not getting why do we need logical operators X-bar and Z-bar

In the image below for stabilizers for shors 9 qubit QECC, I understand the role of S1 to S8 operators but I am not getting role of X-bar and Z-bar operators. I know how to obtain them using standard form but not clear with its use and why do we…
chetan waghmare
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The commutativity of $I$ and $Y$ in a stabilizer code

Let $P_1 = \lbrace I, -I, iI, -iI, X, -X, iX, -iX, Y, -Y, iY, -iY, Z, -Z, iZ, -iZ\rbrace$. Let $P_n$ be the $n$-tensor fold of $P_1$. It is said that two operators either commute if $AB = BA$ or anti-commute if $AB = -BA$ for all $A,B \in P_n$. Let…
M. Al Jumaily
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